Calculus Examples

$V = p r^{2} (2 p r - 24)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d r} (V) = \frac{d}{d r} (p r^{2} (2 p r - 24))$

Step 2

The derivative of $V$ with respect to $r$ is $V'$ .

$V'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $p$ is constant with respect to $r$ , the derivative of $p r^{2} (2 p r - 24)$ with respect to $r$ is $p \frac{d}{d r} [r^{2} (2 p r - 24)]$ .

$p \frac{d}{d r} [r^{2} (2 p r - 24)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d r} [f (r) g (r)]$ is $f (r) \frac{d}{d r} [g (r)] + g (r) \frac{d}{d r} [f (r)]$ where $f (r) = r^{2}$ and $g (r) = 2 p r - 24$ .

$p (r^{2} \frac{d}{d r} [2 p r - 24] + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $2 p r - 24$ with respect to $r$ is $\frac{d}{d r} [2 p r] + \frac{d}{d r} [- 24]$ .

$p (r^{2} (\frac{d}{d r} [2 p r] + \frac{d}{d r} [- 24]) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.2

Since $2 p$ is constant with respect to $r$ , the derivative of $2 p r$ with respect to $r$ is $2 p \frac{d}{d r} [r]$ .

$p (r^{2} (2 p \frac{d}{d r} [r] + \frac{d}{d r} [- 24]) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 1$ .

$p (r^{2} (2 p \cdot 1 + \frac{d}{d r} [- 24]) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.4

Multiply $2$ by $1$ .

$p (r^{2} (2 p + \frac{d}{d r} [- 24]) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.5

Since $- 24$ is constant with respect to $r$ , the derivative of $- 24$ with respect to $r$ is $0$ .

$p (r^{2} (2 p + 0) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $2 p$ and $0$ .

$p (r^{2} (2 p) + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.6.2

Move $2$ to the left of $r^{2}$ .

$p (2 \cdot r^{2} p + (2 p r - 24) \frac{d}{d r} [r^{2}])$

Step 3.3.7

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 2$ .

$p (2 r^{2} p + (2 p r - 24) (2 r))$

Step 3.3.8

Move $2$ to the left of $2 p r - 24$ .

$p (2 r^{2} p + 2 \cdot (2 p r - 24) r)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$p (2 r^{2} p + (2 (2 p r) + 2 \cdot - 24) r)$

Step 3.4.2

Apply the distributive property.

$p (2 r^{2} p + 2 (2 p r) r + 2 \cdot - 24 r)$

Step 3.4.3

Apply the distributive property.

$p (2 r^{2} p) + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4

Combine terms.

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Step 3.4.4.1

Raise $p$ to the power of $1$ .

$p^{1} p (2 r^{2}) + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.2

Raise $p$ to the power of $1$ .

$p^{1} p^{1} (2 r^{2}) + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p^{1 + 1} (2 r^{2}) + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.4

Add $1$ and $1$ .

$p^{2} (2 r^{2}) + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.5

Move $2$ to the left of $p^{2}$ .

$2 \cdot p^{2} r^{2} + p (2 (2 p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.6

Multiply $2$ by $2$ .

$2 p^{2} r^{2} + p (4 (p r) r) + p (2 \cdot - 24 r)$

Step 3.4.4.7

Raise $r$ to the power of $1$ .

$2 p^{2} r^{2} + p (4 p (r^{1} r)) + p (2 \cdot - 24 r)$

Step 3.4.4.8

Raise $r$ to the power of $1$ .

$2 p^{2} r^{2} + p (4 p (r^{1} r^{1})) + p (2 \cdot - 24 r)$

Step 3.4.4.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 p^{2} r^{2} + p (4 p r^{1 + 1}) + p (2 \cdot - 24 r)$

Step 3.4.4.10

Add $1$ and $1$ .

$2 p^{2} r^{2} + p (4 p r^{2}) + p (2 \cdot - 24 r)$

Step 3.4.4.11

Raise $p$ to the power of $1$ .

$2 p^{2} r^{2} + 4 (p^{1} p) r^{2} + p (2 \cdot - 24 r)$

Step 3.4.4.12

Raise $p$ to the power of $1$ .

$2 p^{2} r^{2} + 4 (p^{1} p^{1}) r^{2} + p (2 \cdot - 24 r)$

Step 3.4.4.13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 p^{2} r^{2} + 4 p^{1 + 1} r^{2} + p (2 \cdot - 24 r)$

Step 3.4.4.14

Add $1$ and $1$ .

$2 p^{2} r^{2} + 4 p^{2} r^{2} + p (2 \cdot - 24 r)$

Step 3.4.4.15

Multiply $2$ by $- 24$ .

$2 p^{2} r^{2} + 4 p^{2} r^{2} + p (- 48 r)$

Step 3.4.4.16

Add $2 p^{2} r^{2}$ and $4 p^{2} r^{2}$ .

$6 p^{2} r^{2} + p (- 48 r)$

Step 3.4.5

Reorder terms.

$6 p^{2} r^{2} - 48 p r$

Step 4

Reform the equation by setting the left side equal to the right side.

$V' = 6 p^{2} r^{2} - 48 p r$

Step 5

Replace $V'$ with $\frac{d V}{d r}$ .

$\frac{d V}{d r} = 6 p^{2} r^{2} - 48 p r$